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arXiv:2101.07603 · Physical Review A

Diagrammatic approach to scattering of multiphoton states in waveguide QED

Kiryl Piasotski, Mikhail Pletyukhov

Published Aug 23, 2021• 8 citations
arXivHTMLPDF

Abstract

We give an exposure to diagrammatic techniques in waveguide quantum electrodynamics (QED) systems. A particular emphasis is placed on the systems with delayed coherent quantum feedback. Specifically, we show that the $N$-photon scattering matrices in single-qubit waveguide QED systems, within the rotating wave approximation, admit for a parametrization in terms of ($N\ensuremath{-}1$)-photon effective vertex functions and provide a detailed derivation of a closed hierarchy of generalized Bethe-Salpeter equations satisfied by these vertex functions. The advantage of this method is that the above-mentioned integral equations hold independently of the number of radiation channels, their bandwidth, the dispersion of the modes they are supporting, and the structure of the radiation-qubit coupling interaction, thus enabling one to study multiphoton scattering problems beyond the Born-Markov approximation. Further, we generalize the diagrammatic techniques to the systems containing more than a single emitter by presenting an exact set of equations governing the generic two- and three-photon scattering operators. The above-described theoretical machinery is then showcased on the example of a three-photon scattering on a giant acoustic atom, recently studied experimentally [G. Andersson et al., Nat. Phys. 15, 1123 (2019)].

DOI
10.1103/physreva.104.023709

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We give an exposure to diagrammatic techniques in waveguide quantum electrodynamics (QED) systems. A particular emphasis is placed on the systems with delayed coherent quantum feedback. Specifically, we show that the NNN-photon scattering matrices in single-qubit waveguide QED systems, within the rotating wave approximation, admit for a parametrization in terms of (N\ensuremath−1N\ensuremath{-}1N\ensuremath−1)-photon effective vertex functions and provide a detailed derivation of a closed hierarchy of generalized Bethe-Salpeter equations satisfied by these vertex functions. The advantage of this method is that the above-mentioned integral equations hold independently of the number of radiation channels, their bandwidth, the dispersion of the modes they are supporting, and the structure of the radiation-qubit coupling interaction, thus enabling one to study multiphoton scattering problems beyond the Born-Markov approximation. Further, we generalize the diagrammatic techniques to the systems containing more than a single emitter by presenting an exact set of equations governing the generic two- and three-photon scattering operators. The above-described theoretical machinery is then showcased on the example of a three-photon scattering on a giant acoustic atom, recently studied experimentally [G. Andersson et al., Nat. Phys. 15, 1123 (2019)].
Engineering strong photon–photon interactions at the quantum level have been crucial in various areas of research, notably in quantum information processing and quantum simulation. It is often done by coupling matter strongly to light. A promising way to achieve this is via waveguide quantum electrodynamics (QED). Motivated by these advancements, we study few-photon transport in waveguide QED setups. First, we present a diagrammatic technique to systematically study multiphoton scattering based on the scattering formalism and Green’s function approach. We demonstrate our proposal through physically relevant examples involving scattering of few-photon states from two-level emitters as well as from arrays of correlated Kerr nonlinear resonators described by the Bose–Hubbard model. In the second part, we apply the diagrammatic technique that was developed to perform a comprehensive study on a Bose–Hubbard lattice with a quasi-periodic potential. This model exhibits many-body localisation. We compute the two-photon transmission probability and show that it carries signatures of the underlying localisation transition with close agreement to the participation ratio of the eigenstates. The systematic scattering approach provided in this paper provides a foundation for future works at the interface between quantum optics and condensed matter.
Non-Markovian time evolution of quantum systems is a challenging problem, often mitigated by employing numerical methods or making simplifying assumptions. In this work, we address this problem in waveguide quantum electrodynamics (QED) by developing a diagrammatic approach, which performs fully analytical non-Markovian time evolution of single-photon states. By revisiting Fermi's two-atom problem, we tackle the impeding question of whether the rotating-wave approximation violates causality in single-photon waveguide QED. Afterward, we introduce and prove the no upper half-plane poles (no-UHP) theorem, which connects the poles of scattering parameters to the causality principle. Finally, we visualize the time-delayed coherent quantum feedback mediated by the field and discuss the Markovian limit for microscopically separated qubits where short-distance causality violations occur and the emergence of collective decay rates in this limit. Our diagrammatic approach is a method to perform exact and analytical non-Markovian time evolution of multiemitter systems in waveguide QED.
Waveguide quantum electrodynamics (waveguide QED) describes the interaction between an electromagnetic field confined to a one-dimensional waveguide with atom-like quantum emitters close by. The characteristics of these kinds of systems are the possibility for strong and even ultrastrong interactions between the photonic and atom-like systems, and practically infinite range interactions between the emitters. These properties are valuable for a wide range of quantum optical applications, like quantum communication, quantum networks and quantum metrology. In this thesis we focus on two challenges in quantum optics, namely the generation and scattering of multiphoton states. In the first part of this thesis we demonstrate how waveguide QED systems can be exploited for the generation of multiphoton states, in particular of single-mode Fock states and superpositions thereof as well as multi-mode photonic states with metrological applications. The basic setup for this goal is an ensemble of quantum emitters coupled to a waveguide. In the so-called atomic mirror configuration symmetric Dicke states (or a superposition thereof) decay superradiantly to the ground state and emit the desired multiphoton state, which can be efficiently collected at the ends of the waveguide. We propose various protocols for the preparation of these Dicke states in waveguide QED systems and characterize the emitted photonic state. It turns out that in the low excitation regime, that is, if the number of photons is much lower than the ensemble size, the photonic state is emitted into a single mode. This single-mode structure is fundamental to current proposals for applications in quantum metrology with optical interferometers. Outside the low excitation regime, a multi-mode photonic state is generated, for which the metrological capabilities were unknown. We were able to show that these states still lead to quantum-enhanced optical interferometry. In the second part of this thesis we demonstrate how the scattering of multiphoton states on a single quantum emitter coupled to the waveguide can be used for testing the limits of the light-matter interaction strength. In particular, we investigate systems in the so-called ultrastrong coupling regime, where the coupling strength between the photonic and atom-like system is of the order of the emitter's transition energy. In this regime, many methods and approximations used in quantum optics, especially the Rotating Wave Approximation, break down and one needs to develop new analytical and numerical methods to study these systems. One approach is the so-called polaron Transformation. We show how this transformation can be applied for predicting the scattering amplitude of few photons scattering on the emitter. The comparison of these results with numerical simulations using matrix product states shows a good agreement for moderate coupling strengths. Both parts together show that waveguide QED systems are promising candidates for the generation of multiphoton states as well as for the investigation of fundamental limits probed through the scattering of multiphoton states. The latter also finds application in the implementation of photon-photon nonlinearities induced by the interaction with the quantum emitters. Apart from these and many other theoretical predictions, waveguide QED systems are also undergoing rapid progress in experiments. Therefore, we expect these kinds of systems to bring forth several advances in the field of quantum optics.
We investigate the problem of arbitrary photonic-Fock-state scattering in a waveguide-QED system which consists of a one-dimensional waveguide coupled to a two-level system. By imposing the open boundary conditions that directly describe the physical settings, we construct a complete set of eigenstates of the system, and elucidate the mathematical structures of the eigenstates. In particular, we show that the eigenstates include a set of multiphoton extended states and multiphoton bound states, formed by all possible partitions of the photon number NNN. The total number of the eigenstates is exactly described by the integer number partition function Z(N)Z(N)Z(N). Using the set of eigenstates, we form the scattering matrix, which facilitates the calculations of the scattered photon states, for the scattering processes. With the scattered photon states, we compute the photon correlation functions that manifestly exhibit the bunching and antibunching behaviors in the scattered photon states. As a concrete example, we discuss in detail with a focus on the three-photon Fock state. Such a capability to generate photonic entanglement from unentangled Fock states will have broad applications in quantum information processing.
We present a quantum dynamical study of pulsed few-photon scattering from a single artificial atom, consisting of a two-level system (TLS) or qubit, in a waveguide QED system, directly comparing and contrasting two different quantum theoretical simulation methods: (i) an input-output scattering approach that uses frequency-dependent scattering matrices, and (ii) a matrix product states (MPS) approach, which uses quantum noise operators in time bins and a tensor network technique to solve the time-dependent waveguide function for the entire system. Beginning with pulsed excitation using one-photon and two-photon Fock state pulses, we first show how to compute time-dependent observables with the scattering matrix approach, in terms of frequency integrals that encode the pulse spectrum, including how to extract the population dynamics of the excited quantum emitter, as well as the linear and nonlinear contributions. We present solutions for both symmetric and chiral TLS coupling. We then show how to compute the qubit and field observables in a more direct way using MPS, and obtain the characteristic bird-like shape for the two-photon correlation function at two times, which has been observed in recent experiments. We compare and contrast both of these methods, for one and two-photon excitation pulses, and show excellent agreement. We also present a study of the linear and nonlinear contributions, which can easily be calculated using scattering theory, and show the important role of pulse duration. Finally, we demonstrate the clear advantages of MPS by easily going to higher N-photon excitations, and show selected example population dynamics of up to eight-photon Fock-state pulses, manifesting in clear nonlinear population oscillations during the pulse interaction, similar to classical Rabi oscillations, but with quantum input fields that have a vanishing electric field expectation value.
Single-photon scattering in waveguide-QED systems plays a pivotal role in quantum information processing and communication. We introduce a time-dependent variational approach employing a modified multiple Davydov Ansatz to simulate single-photon dynamics in one-dimensional waveguide-QED systems within the strong-coupling regime. The method is benchmarked across three configurations: a single qubit, two distant qubits, and a ladder-type qutrit coupled to the waveguide. For the single- and two-qubit cases, the variational results exhibit excellent agreement with those derived from a transformed rotating-wave approximation (TRWA). The accuracy of our approach is further verified by an analysis of the local-in-time error, which remains small even under strong coupling, indicating its potential superiority over TRWA in specific regimes. In addition, by contrasting our results with the rotating-wave approximation, we elucidate the significant influence of counter-rotating couplings on scattering dynamics. Our findings establish this variational method as a reliable and versatile tool for investigating single-photon scattering in diverse waveguide-QED systems, particularly in regimes where traditional approximations falter.